I'm just reading through the definition of a ring and it says that "A ring is the triple (R,+,•) where R is a set and +,• are binary operations".

Does • imply multiplication since for R to be a ring we require associativity of multiplication and the existence of a multiplicative inverse?

Or can it be another binary operation yet those two conditions still hold (despite multiplication not being a binary operation)?

Thank you!

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    $\begingroup$ +,• are just suggestive symbols for these binary operations, you might as well use a square and a blue triangle. The important thing is that these binary operations have the properties postulatd thereafter $\endgroup$ – Hagen von Eitzen May 6 '18 at 7:36
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    $\begingroup$ Incidentally, we don't require the existence of multiplicative inverses in a ring. $\endgroup$ – Eric Wofsey May 6 '18 at 7:36

The operations $+$ and $\cdot$ can be any binary operations on the set $R$, as long as they satisfy all the ring axioms. They don't have to be the operations we normally call "addition" and "multiplication" on $R$, if such operations exist. However, in the context of the ring $(R,+,\cdot)$, we usually refer to $+$ as "addition" and $\cdot$ as "multiplication". So when we speak of "associativity of multiplication", for instance, we are actually talking about associativity of the operation $\cdot$, whatever it happens to be.

  • $\begingroup$ Ahhh! okay thank you! I was under the impression the addition was a must (in terms of how we think about it in the reals) - that was just my mistake for misinterpreting my lecture notes. Thank you very much! $\endgroup$ – BigWig May 6 '18 at 7:40

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